约束优化进化算法

约束优化问题是科学和工程应用领域经常会遇到的一类数学规划问题.近年来,约束优化问题求解已成为进化计算研究的一个重要方向.从约束优化进化算法=约束处理技术+进化算法的研究框架出发,从约束处理技术和进化算法两个基本方面对约束优化进化算法的研究及进展进行了综述.此外,对约束优化进化算法中的一些重要问题进行了探讨.最后进行了各种算法的比较性总结,深入分析了目前约束优化进化算法中亟待解决的问题,并指出了值得进一步研究的方向.     

An adaptive penalty based covariance matrix adaptation–evolution strategy

Although most of unconstrained optimization problems with moderate to high dimensions can be easily handled with Evolutionary Computation (EC) techniques, constraint optimization problems (COPs) with inequality and equality constraints are very hard to deal with. Despite the fact that only equality constraints can be used to eliminate a certain variable, both types of constraints implicitly enforce a relation between problem variables. Most conventional constraint handling methods in EC do not consider the correlations between problem variables imposed by the problem constraints. This paper relies on the idea that a proper genetic operator, which captures mentioned implicit correlations, can improve performance of evolutionary constrained optimization algorithms. With this in mind, we employ a (μ+λ)-Evolution Strategy with a simplified variant of Covariance Matrix Adaptation based mutation operator along an adaptive weight adjustment scheme. The proposed algorithm is tested on two test sets. The outperformance of the algorithm is significant on the first benchmark when compared with five conventional methods. The results on the second test set show that algorithm is highly competitive when benchmarked with three state-of-art algorithms. The main drawback of the algorithm is its slightly lower speed of convergence for problems with high dimension and/or large search domain.     

Using variable reduction strategy to accelerate evolutionary optimization

In this study, we introduce a novel approach of variable reduction and integrate it into evolutionary algorithms in order to reduce the complexity of optimization problems. We develop reduction processes of variable reduction for derivative unconstrained optimization problems (DUOPs) and constrained optimization problems (COPs) with equality constraints and active inequality constraints. Variable reduction uses the problem domain knowledge implied when investigating optimal conditions existing in optimization problems. For DUOPs, equations involving derivatives are considered while for COPs, we discuss equations expressing the equality constraints. From the relationships formed in this way, we obtain relationships among the variables that have to be satisfied by optimal solutions. According to such relationships, we can utilize some variables (referred to as core variables) to express some other variables (referred to as reduced variables). We show that the essence of variable reduction is to produce a minimum collection of core variables and a maximum number of reduced variables based on a system of equations. We summarize some application-oriented situations of variable reduction and stress several important issues related to the further application and development of variable reduction. Essentially, variable reduction can reduce the number of variables and eliminate equality constraints, thus reducing the dimensionality of the solution space and improving the efficiency of evolutionary algorithms. The approach can be applied to unconstrained, constrained, continuous and discrete optimization problems only if there are explicit variable relationships to be satisfied in the optimal conditions. We test variable reduction on real-world and synthesized DUOPs and COPs. Experimental results and comparative studies point at the effectiveness of variable reduction.     

面向复杂约束优化问题的进化算法综述

约束优化是多数实际工程应用优化问题的呈现方式.进化算法由于其高效的表现,近年来被广泛应用于约束优化问题求解.但约束条件使得问题解空间离散、缩小、改变,给进化算法求解约束优化问题带来极大挑战.在此背景下,融合约束处理技术的进化算法成为研究热点.此外,随着研究的深入,近年来约束处理技术在复杂工程应用问题优化中得到了广泛发展,例如多目标、高维、等式优化等.根据复杂性的缘由,将面向复杂约束优化问题的进化优化分为面向复杂目标的进化约束优化算法和面向复杂约束场景的进化算法两种类别进行综述,其中,重点探讨了实际工程应用的复杂性对约束处理技术的挑战和目前研究的最新进展,并最后总结了未来的研究趋势与挑战.     

基于内部罚函数的进化算法求解约束优化问题

为解决现有约束处理方法可行解的适应度函数不包含约束条件的问题,提出了一种内部罚函数候选解筛选规则.该候选解筛选规则分别对可行解和不可行解采用内部罚函数和约束违反度进行筛选,从而达到平衡最小化目标函数和满足约束条件的目的.以进化策略算法为基础,给出了基于内部罚函数候选解筛选规则的进化算法的一个实现.进一步地,从理论和实验角度分别验证了内部罚函数候选解筛选规则的有效性:以(1+1)进化算法为例,从进化成功率方面验证了内部罚函数候选解筛选规则的理论有效性;通过13个测试问题的数值实验,从进化成功率、候选解后代是可行解的比例、进化步长和收敛速度方面验证了内部罚函数候选解筛选规则的实验有效性.     

Biogeography-based optimization for constrained optimization problems

Biogeography-based optimization (BBO) has been recently proposed as a viable stochastic optimization algorithm and it has so far been successfully applied in a variety of fields, especially for unconstrained optimization problems. The present paper shows how BBO can be applied for constrained optimization problems, where the objective is to find a solution for a given objective function, subject to both inequality and equality constraints.     

A simple and efficient real-coded genetic algorithm for constrained optimization

This paper presents a simple and efficient real-coded genetic algorithm (RCGA) for constrained real-parameter optimization. Different from some conventional RCGAs that operate evolutionary operators in a series framework, the proposed RCGA implements three specially designed evolutionary operators, named the ranking selection (RS), direction-based crossover (DBX), and the dynamic random mutation (DRM), to mimic a specific evolutionary process that has a parallel-structured inner loop. A variety of benchmark constrained optimization problems (COPs) are used to evaluate the effectiveness and the applicability of the proposed RCGA. Besides, some existing state-of-the-art optimization algorithms in the same category of the proposed algorithm are considered and utilized as a rigorous base of performance evaluation. Extensive comparison results reveal that the proposed RCGA is superior to most of the comparison algorithms in providing a much faster convergence speed as well as a better solution accuracy, especially for problems subject to stringent equality constraints. Finally, as a specific application, the proposed RCGA is applied to optimize the GaAs film growth of a horizontal metal-organic chemical vapor deposition reactor. Simulation studies have confirmed the superior performance of the proposed RCGA in solving COPs.     

求解约束优化问题的协同进化教与学优化算法

对约束优化问题,为了避免罚因子和等式约束转化为不等式约束时引入的约束容忍度参数所带来的不便,本文在基本教与学优化（Teaching-learning-based optimization,TLBO）算法中加入了自我学习过程并提出了一种求解约束优化问题的协同进化教与学优化算法,使得罚因子和约束容忍度随种群的进化动态调整.对7个常见测试函数的数值实验验证了算法求解带有等式和不等式约束优化问题的有效性.     

Extension of a hybrid Genetic Algorithm for nonlinear programming problems with equality and inequality constraints

As an extension of the hybrid Genetic Algorithm-HGA proposed by Tang et al. (Comput. Math. Appl. 36 (1998) 11), this paper focuses on the critical techniques in the application of the GA to nonlinear programming (NLP) problems with equality and inequality constraints. Taking into account the equality constraints and embedding the information of infeasible points/chromosomes into the evaluation function, an extended fuzzy-based methodology and three new evaluation functions are proposed to formulate and evaluate the infeasible chromosomes. The extended version of concepts of dominated semi-feasible direction (DSFD), feasibility degree (FD₁) of semi-feasible direction, feasibility degree (FD₂) of infeasible points ‘belonging to’ feasible domain are introduced. Combining the new evaluation functions and weighted gradient direction search into the Genetic Algorithm, an extended hybrid Genetic Algorithm (EHGA) is developed to solve nonlinear programming (NLP) problems with equality and inequality constraints. Simulation shows that this new algorithm is efficient.Scope and purposeNon-linear Programming (NLP) problems with equality and inequality constraints is an important type of constrained optimization problems. Genetic Algorithm (GA) is one of the well known evolutionary computation techniques. In the application of GA to NLP problems, chromosomes randomly generated at the beginning and/or generated by genetic operators during the evolutionary process usually violate the constraints, resulting in infeasible chromosomes. Therefore, the handling of system constraints, particularly the nonlinear equation constraints, and the measurement and evaluation of infeasible chromosomes, are major concerns in GA. Penalty strategy in the construction of fitness function is commonly used to evaluate the infeasible chromosomes in some traditional AG methods. However, this approach essentially narrows down the search space by eliminating all infeasible chromosomes from the evolutionary process, and it may reduce the chances of finding better candidates for the global optimization. In particular, it absolutely ignores the information carried by the infeasible chromosomes itself. Therefore, formulating the infeasible chromosomes by embedding the relevant information into the evaluation function are important when applying GA to NLP.As an extension of the Hybrid Genetic Algorithm-HGA proposed by Tang et al. (1998), this paper focuses on the critical techniques in the application of GA to NLP problems with equality and inequality constraints. Taking into account the equality constraints and embedding the information of infeasible chromosomes into the evaluation function, an extended fuzzy-based methodology and three new evaluation functions are designed to formulate and evaluate the infeasible chromosomes. By introducing an extended version of the concepts of dominated semi-feasible direction (DSFD), feasibility degree (FD₁) of semi-feasible direction, feasibility degree (FD₂) of infeasible points ‘belonging to’ feasible domain, an extended hybrid Genetic Algorithm (EHGA) is developed for solving NLP problems with equality and inequality constraints.     

一种新的遗传算法求解有等式约束的优化问题

针对有等式约束的优化问题,提出了一种新的遗传算法.该算法是在种群初始化、交叉、变异操作过程中使用求解参数方程的方法处理等式约束,违反不等式约束的个体用死亡罚函数进行惩罚设计出的实数编码遗传算法.数值实验结果表明,新算法性能优于现有其它算法;它不仅可以处理线性等式约束,而且还可以处理非线性等式约束,同时提高了收敛速度和解的精度,是一种通用强、高效稳健的智能算法.     

基于梯度投影法与随机优化算法的约束优化方法

针对带有线性等式和不等式约束的无确定函数形式的约束优化问题,提出一种利用梯度投影法与遗传算法、同时扰动随机逼近等随机算法相结合的优化方法。该方法利用遗传算法进行全局搜索,利用同时扰动随机逼近算法进行局部搜索,算法在每次进化时根据线性约束计算父个体处的梯度投影方向,以产生新个体,从而能够严格保证新个体满足全部约束条件。将上述约束优化算法应用于典型约束优化问题,其仿真结果表明了所提出算法的可行性和收敛性。     

An efficient multiobjective differential evolution algorithm for engineering design

Solving engineering design and resources optimization via multiobjective evolutionary algorithms (MOEAs) has attracted much attention in the last few years. In this paper, an efficient multiobjective differential evolution algorithm is presented for engineering design. Our proposed approach adopts the orthogonal design method with quantization technique to generate the initial archive and evolutionary population. An archive (or secondary population) is employed to keep the nondominated solutions found and it is updated by a new relaxed form of Pareto dominance, called Pareto-adaptive ϵ-dominance (paϵ-dominance), at each generation. In addition, in order to guarantee to be the best performance produced, we propose a new hybrid selection mechanism to allow the archive solutions to take part in the generating process. To handle the constraints, a new constraint-handling method is employed, which does not need any parameters to be tuned for constraint handling. The proposed approach is tested on seven benchmark constrained problems to illustrate the capabilities of the algorithm in handling mathematically complex problems. Furthermore, four well-studied engineering design optimization problems are solved to illustrate the efficiency and applicability of the algorithm for multiobjective design optimization. Compared with Nondominated Sorting Genetic Algorithm II, one of the best MOEAs available at present, the results demonstrate that our approach is found to be statistically competitive. Moreover, the proposed approach is very efficient and is capable of yielding a wide spread of solutions with good coverage and convergence to true Pareto-optimal fronts.     

Using augmented Lagrangian particle swarm optimization for constrained problems in engineering">Using augmented Lagrangian particle swarm optimization for constrained problems in engineering

The comparatively new stochastic method of particle swarm optimization (PSO) has been applied to engineering problems especially of nonlinear, non-differentiable, or non-convex type. Its robustness and its simple applicability without the need for cumbersome derivative calculations make PSO an attractive optimization method. However, engineering optimization tasks often consist of problem immanent equality and inequality constraints which are usually included by inadequate penalty functions when using stochastic algorithms. The simple structure of basic particle swarm optimization characterized by only a few lines of computer code allows an efficient implementation of a more sophisticated treatment of such constraints. In this paper, we present an approach which utilizes the simple structure of the basic PSO technique and combines it with an extended non-stationary penalty function approach, called augmented Lagrange multiplier method, for constraint handling where ill conditioning is a far less harmful problem and the correct solution can be obtained even for finite penalty factors. We describe the basic PSO algorithm and the resulting method for constrained problems as well as the results from benchmark tests. An example of a stiffness optimization of an industrial hexapod robot with parallel kinematics concludes this paper and shows the applicability of the proposed augmented Lagrange particle swarm optimization to engineering problems.     

Constrained differential evolution using generalized opposition-based learning

Differential evolution (DE) is a well-known optimization approach to deal with nonlinear and complex optimization problems. However, many real-world optimization problems are constrained problems that involve equality and inequality constraints. DE with constraint handling techniques, named constrained differential evolution (CDE), can be used to solve constrained optimization problems. In this paper, we propose a new CDE framework that uses generalized opposition-based learning (GOBL), named GOBL-CDE. In GOBL-CDE, firstly, the transformed population is generated using general opposition-based learning in the population initialization. Secondly, the transformed population and the initial population are merged and only half of the best individuals are selected to compose the new initial population to proceed mutation, crossover, and selection. Lastly, based on a jumping probability, the transformed population is calculated again after generating new populations, and the fittest individuals are selected to compose new population from the union of the current population and the transformed population. The GOBL-CDE framework can be applied to most CDE variants. As examples, in this study, the framework is applied to two popular representative CDE variants, i.e., rank-iMDDE and \(\varepsilon \)DEag. Experiment results on 24 benchmark functions from CEC’2006 and 18 benchmark functions from CEC’2010 show that the proposed framework is an effective approach to enhance the performance of CDE algorithms.     

Distributed continuous‐time algorithm for a general nonsmooth monotropic optimization problem

This paper investigates a general monotropic optimization problem for continuous‐time networks, where the global objective function is a sum of local objective functions that are only known to individual agent, and general constraints are taken into account, including local inequality constraints, global equality constraint, and local feasible constraints. In addition, all functions involved in the objective functions and inequality constraints are not necessarily differentiable. To solve the problem, a distributed continuous‐time algorithm is designed using subgradient projections, and it is shown that the proposed algorithm is well defined in the sense that the existence of its solutions can be guaranteed. Furthermore, it is proved that the algorithm converges to an optimal solution for the general monotropic optimization problem. Finally, a simulation example is provided for validating the theoretical result.     

求解约束优化的带偏好双目标优化方法

文章提出了一种求解约束优化问题的新方法。它把约束优化问题转化为双目标优化问题,一个目标是原问题的目标,另一目标是由约束条件转化得到。转化得到的双目标优化与一般的双目标优化问题不同在于它偏好那些使约束条件满足的最优解。我们利用动态权值将这一带有偏好的双目标优化转化为无约束的单目标优化,并使其满足偏好特性。我们对四个标准测试函数进行了数据仿真实验,实验结果表明该算法是有效的。     

Multiobjective optimization and hybrid evolutionary algorithm to solve constrained optimization problems.

This paper presents a novel evolutionary algorithm (EA) for constrained optimization problems, i.e., the hybrid constrained optimization EA (HCOEA). This algorithm effectively combines multiobjective optimization with global and local search models. In performing the global search, a niching genetic algorithm based on tournament selection is proposed. Also, HCOEA has adopted a parallel local search operator that implements a clustering partition of the population and multiparent crossover to generate the offspring population. Then, nondominated individuals in the offspring population are used to replace the dominated individuals in the parent population. Meanwhile, the best infeasible individual replacement scheme is devised for the purpose of rapidly guiding the population toward the feasible region of the search space. During the evolutionary process, the global search model effectively promotes high population diversity, and the local search model remarkably accelerates the convergence speed. HCOEA is tested on 13 well-known benchmark functions, and the experimental results suggest that it is more robust and efficient than other state-of-the-art algorithms from the literature in terms of the selected performance metrics, such as the best, median, mean, and worst objective function values and the standard deviations.     

基于连续时间的二阶多智能体分布式资源分配算法

针对二阶多智能体系统中的分布式资源分配问题, 本文设计两种连续时间算法. 基于KKT (Karush?Kuhn?Tucker, 卡罗需?库恩?塔克)优化条件, 第一种控制算法利用节点局部不等式及其梯度信息来约束节点状态. 与上述梯度方法不同, 第二种控制算法包括一致性梯度下降法和固定时间收敛映射算子, 其中固定时间收敛映射算子确保算法的节点状态在固定时间收敛到局部约束集, 一致性梯度下降法目的是确保节点迭代到资源分配问题最优解. 两种控制算法都对状态无初始值约束, 且控制参数都是常数. 利用凸优化理论和固定时间李雅普诺夫方法, 分别分析了上述控制策略在有向平衡网络条件下的渐近和指数收敛性. 最后通过数值仿真验证了所设计算法在一维和高维资源分配问题的有效性.     

Practical application of a penalty function approach to constrained minimax optimization

A practical, penalty function approach to solving constrained minimax problems is applied here. In essence, this approach reformulates the constrained minimax problem as an unconstrained minimax problem. A recently proposed optimization algorithm called grazor search is used to solve the reformulated unconstrained minimax problem. The proposed approach can handle inequality constraints-parameter constraints in particular. A practical transmission-line filter example with parameter constraints illustrates the results.     

A hybrid differential evolution algorithm based on particle swarm optimization for nonconvex economic dispatch problems

This paper presents the design and application of an efficient hybrid heuristic search method to solve the practical economic dispatch problem considering many nonlinear characteristics of power generators, and their operational constraints, such as transmission losses, valve-point effects, multi-fuel options, prohibited operating zones, ramp rate limits and spinning reserve. These practical operation constraints which can usually be found at the same time in realistic power system operations make the economic load dispatch problem a nonsmooth optimization problem having complex and nonconvex features with heavy equality and inequality constraints.The proposed approach combines in the most effective way the properties of two of the most popular evolutionary optimization techniques now in use for power system optimization, the Differential Evolution (DE) and Particle Swarm Optimization (PSO) algorithms. To improve the global optimization property of DE, the PSO procedure is integrated as additional mutation operator.The effectiveness of the proposed algorithm (termed DEPSO) is demonstrated by solving four kinds of ELD problems with nonsmooth and nonconvex solution spaces. The comparative results with some of the most recently published methods confirm the effectiveness of the proposed strategy to find accurate and feasible optimal solutions for practical ELD problems.